Date Available
4-20-2018
Year of Publication
2018
Document Type
Doctoral Dissertation
Degree Name
Doctor of Philosophy (PhD)
College
Arts and Sciences
Department/School/Program
Mathematics
Advisor
Dr. Benjamin J. Braun
Abstract
The broad topic of this dissertation is the study of algebraic structure arising from polyhedral geometric objects. There are three distinct topics covered over three main chapters. However, each of these topics are further linked by a connection to the Eulerian polynomials.
Chapter 2 studies Euler-Mahonian identities arising from both the symmetric group and generalized permutation groups. Specifically, we study the algebraic structure of unit cube semigroup algebra using Grรถbner basis methods to acquire these identities. Moreover, this serves as a bridge between previous methods involving polyhedral geometry and triangulations with descent bases methods arising in representation theory.
In Chapter 3, the aim is to characterize Hilbert basis elements of certain ๐-lecture hall cones. In particular, the main focus is the classification of the Hilbert bases for the 1 mod ๐ cones and the ๐-sequence cones, both of which generalize a previous known result. Additionally, there is much broader characterization of Hilbert bases in dimension โค 4 for ๐-generated Gorenstein lecture hall cones.
Finally, Chapter 4 focuses on certain algebraic and geometric properties of ๐-lecture hall polytopes. This consists of partial classification results for the Gorenstein property, the integer-decomposition property, and the existence of regular, unimodular triangulations.
Digital Object Identifier (DOI)
https://doi.org/10.13023/ETD.2018.144
Recommended Citation
Olsen, McCabe J., "HILBERT BASES, DESCENT STATISTICS, AND COMBINATORIAL SEMIGROUP ALGEBRAS" (2018). Theses and Dissertations--Mathematics. 52.
https://uknowledge.uky.edu/math_etds/52
